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# Matrix Approach to Simple Linear Regression

Use of Inverse

Consider equation • Inverse similar to using reciprocal of a scalar

• Pertains to a set of equations • Assuming A has an inverse: Random Vectors and Matrices

• Contain elements that are random variables

• Can compute expectation and (co)variance

• In regression set up, both ε and Y are random
vectors

• Expectation vector: • Covariance matrix: symmetric Basic Theorems
• Consider random vector Y

• Consider constant matrix A

• Suppose W = AY
– W is also a random vector
– E(W) = AE(Y) Regression Matrices

• Can express observations
Y = Xβ +ε

• Both Y and ε are random vectors

E(Y)
= Xβ+E(ε)
= Xβ Least Squares

• Express quantity Q • Taking derivative → • This means Fitted Values

• The fitted values • Matrix is called the hat matrix
– H is symmetric, i.e., H' = H
– H is idempotent, i.e., HH = H

Equivalently write • Matrix H used in diagnostics (chapter 9)

Residuals

• Residual matrix • e a random vector ANOVA where A is symmetric n × n matrix

Sums of squares can be shown to be quadratic forms (page
207)

Quadratic forms play significant role in the theory of linear
models when errors are normally distributed

Inference

• Vector • The mean and variance are • Thus, b is multivariate Normal • Consider • Mean response  Prediction of new observation Chapter Review

• Review of Matrices

• Regression Model in Matrix Form

Calculations Using Matrices

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